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Volumn 62, Issue 1, 2007, Pages 1-43

On organizing principles of discrete differential geometry. Geometry of spheres

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EID: 34547411513     PISSN: 00360279     EISSN: 14684829     Source Type: Journal    
DOI: 10.1070/RM2007v062n01ABEH004380     Document Type: Article
Times cited : (39)

References (76)
  • 1
    • 85164645298 scopus 로고    scopus 로고
    • A.I. Bobenko and P. Schröder, Discrete Willmore flow, Eurographics Symposium on Geometry Processing (Vienna, 2005) (M. Desbrun and H. Pottmann, eds.), Eurographics Assoc., Aire-la-Ville, Switzerland 2005, pp. 101-110.
    • A.I. Bobenko and P. Schröder, "Discrete Willmore flow", Eurographics Symposium on Geometry Processing (Vienna, 2005) (M. Desbrun and H. Pottmann, eds.), Eurographics Assoc., Aire-la-Ville, Switzerland 2005, pp. 101-110.
  • 2
    • 33749258553 scopus 로고    scopus 로고
    • Geometric modeling with conical meshes and developable surfaces
    • Y. Liu, H. Pottmann, J. Wallner, Y.-L. Yang, and W. Wang, "Geometric modeling with conical meshes and developable surfaces", ACM Trans. Graphics 25:3 (2006), 681-689.
    • (2006) ACM Trans. Graphics , vol.25 , Issue.3 , pp. 681-689
    • Liu, Y.1    Pottmann, H.2    Wallner, J.3    Yang, Y.-L.4    Wang, W.5
  • 3
    • 85164635188 scopus 로고    scopus 로고
    • Russian source
    • Russian source
  • 4
    • 85164644869 scopus 로고    scopus 로고
    • English transl., A.D. Alexandrov, Convex polyhedra, Springer Monogr. Math., Springer-Verlag, Berlin 2005.
    • English transl., A.D. Alexandrov, Convex polyhedra, Springer Monogr. Math., Springer-Verlag, Berlin 2005.
  • 5
    • 85164645259 scopus 로고    scopus 로고
    • Russian source
    • Russian source
  • 6
    • 85164657320 scopus 로고    scopus 로고
    • English transl., A. V. Pogorelov, Extrinsic geometry of convex surfaces, Transl. Math. Monogr., 35, Amer. Math. Soc., Providence, RI 1973.
    • English transl., A. V. Pogorelov, Extrinsic geometry of convex surfaces, Transl. Math. Monogr., vol. 35, Amer. Math. Soc., Providence, RI 1973.
  • 7
    • 0008982002 scopus 로고
    • Parallelogrammgitter als Modelle pseudosphärischer Flächen
    • R. Sauer, "Parallelogrammgitter als Modelle pseudosphärischer Flächen", Math. Z. 52 (1950), 611-622.
    • (1950) Math. Z , vol.52 , pp. 611-622
    • Sauer, R.1
  • 8
    • 0009079135 scopus 로고
    • Zur Differenzengeometrie der Flächen konstanter negativer Krümmung
    • IIa
    • W. Wunderlich, "Zur Differenzengeometrie der Flächen konstanter negativer Krümmung", Österreich. Akad. Wiss. Math.-Nat. Kl. S.-B. IIa 160 (1951), 39-77.
    • (1951) Österreich. Akad. Wiss. Math.-Nat. Kl. S.-B , vol.160 , pp. 39-77
    • Wunderlich, W.1
  • 10
    • 21344457887 scopus 로고    scopus 로고
    • Harmonic functions for quadrilateral remeshing of arbitrary manifolds
    • S. Dong, S. Kircher, and M. Garland, "Harmonic functions for quadrilateral remeshing of arbitrary manifolds", Comput. Aided Geom. Design 22:5 (2005), 392-423.
    • (2005) Comput. Aided Geom. Design , vol.22 , Issue.5 , pp. 392-423
    • Dong, S.1    Kircher, S.2    Garland, M.3
  • 13
    • 0002821692 scopus 로고    scopus 로고
    • Discrete surfaces with constant negative Gaussian curvature and the Hirota equation
    • A.I. Bobenko and U. Pinkall, "Discrete surfaces with constant negative Gaussian curvature and the Hirota equation", J. Differential Geom. 43:3 (1996), 527-611.
    • (1996) J. Differential Geom , vol.43 , Issue.3 , pp. 527-611
    • Bobenko, A.I.1    Pinkall, U.2
  • 14
  • 15
    • 0000894684 scopus 로고    scopus 로고
    • Multidimensional quadrilateral lattices are integrable
    • A. Doliwa and P.M. Santini, "Multidimensional quadrilateral lattices are integrable", Phys. Lett. A 233:4-6 (1997), 365-372.
    • (1997) Phys. Lett. A , vol.233 , Issue.4-6 , pp. 365-372
    • Doliwa, A.1    Santini, P.M.2
  • 16
    • 0039245205 scopus 로고    scopus 로고
    • The integrable discrete analogues of orthogonal coordinate systems are multi-dimensional circular lattices
    • J. Cieśliński, A. Doliwa, and P. M. Santini, "The integrable discrete analogues of orthogonal coordinate systems are multi-dimensional circular lattices", Phys. Lett. A 235:5 (1997), 480-488.
    • (1997) Phys. Lett. A , vol.235 , Issue.5 , pp. 480-488
    • Cieśliński, J.1    Doliwa, A.2    Santini, P.M.3
  • 17
    • 85164634937 scopus 로고    scopus 로고
    • Russian source
    • Russian source
  • 18
    • 0031351831 scopus 로고    scopus 로고
    • Discrete spectral symmetries of low-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds
    • English transl., S. P. Novikov and I. A. Dynnikov, "Discrete spectral symmetries of low-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds", Russian Math. Surveys 52:5 (1997), 1057-1116.
    • (1997) Russian Math. Surveys , vol.52 , Issue.5 , pp. 1057-1116
    • English transl, S.1    Novikov, P.2    Dynnikov, I.A.3
  • 19
    • 0001204271 scopus 로고    scopus 로고
    • Schrödinger operators on graphs and symplectic geometry, The Arnoldfest (Toronto, ON, 1997), Fields Inst. Commun., vol. 24
    • S.P. Novikov, "Schrödinger operators on graphs and symplectic geometry", The Arnoldfest (Toronto, ON, 1997), Fields Inst. Commun., vol. 24, Amer. Math. Soc., Providence, RI 1999, pp. 397-413.
    • Amer. Math. Soc., Providence , vol.RI 1999 , pp. 397-413
    • Novikov, S.P.1
  • 20
    • 85164650246 scopus 로고    scopus 로고
    • Russian source
    • Russian source
  • 22
    • 33644804677 scopus 로고    scopus 로고
    • Geometry of the triangle equation on two-manifolds
    • I. A. Dynnikov and S. P. Novikov, "Geometry of the triangle equation on two-manifolds", Mosc. Math. J. 3:2 (2003), 419-438.
    • (2003) Mosc. Math. J , vol.3 , Issue.2 , pp. 419-438
    • Dynnikov, I.A.1    Novikov, S.P.2
  • 23
    • 85164648704 scopus 로고    scopus 로고
    • A. I. Bobenko and U. Pinkall, Discretization of surfaces and integrable systems, Discrete integrable geometry and physics (Vienna, 1996) (A.I. Bobenko and R. Seiler, eds.), Oxford Lecture Ser. Math. Appl., 16, Oxford Univ. Press, New York 1999, pp. 3-58.
    • A. I. Bobenko and U. Pinkall, "Discretization of surfaces and integrable systems", Discrete integrable geometry and physics (Vienna, 1996) (A.I. Bobenko and R. Seiler, eds.), Oxford Lecture Ser. Math. Appl., vol. 16, Oxford Univ. Press, New York 1999, pp. 3-58.
  • 25
    • 0036373495 scopus 로고    scopus 로고
    • Integrable systems on quad-graphs
    • A.I. Bobenko and Yu. B. Suris, "Integrable systems on quad-graphs", Int. Math. Res. Not. 11 (2002), 573-611.
    • (2002) Int. Math. Res. Not , vol.11 , pp. 573-611
    • Bobenko, A.I.1    Suris, Y.B.2
  • 26
    • 0037330605 scopus 로고    scopus 로고
    • Classification of integrable equations on quad-graphs. The consistency approach
    • V. E. Adler, A. I. Bobenko, and Yu. B. Suris, "Classification of integrable equations on quad-graphs. The consistency approach", Comm. Math. Phys. 233:3 (2003), 513-543.
    • (2003) Comm. Math. Phys , vol.233 , Issue.3 , pp. 513-543
    • Adler, V.E.1    Bobenko, A.I.2    Suris, Y.B.3
  • 27
    • 0033130159 scopus 로고    scopus 로고
    • Quadratic reductions of quadrilateral lattices
    • A. Doliwa, "Quadratic reductions of quadrilateral lattices", J. Geom. Phys. 30:2 (1999), 169-186.
    • (1999) J. Geom. Phys , vol.30 , Issue.2 , pp. 169-186
    • Doliwa, A.1
  • 28
    • 4243350830 scopus 로고
    • Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie
    • Springer-Verlag, Berlin
    • W. Blaschke, Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. III: Differentialgeometrie der Kreise und Kugeln, Springer-Verlag, Berlin 1929.
    • (1929) III: Differentialgeometrie der Kreise und Kugeln
    • Blaschke, W.1
  • 29
    • 0004265477 scopus 로고
    • Oxford Sci. Publ, Oxford Univ. Press, New York
    • T. J. Willmore, Riemannian geometry, Oxford Sci. Publ., Oxford Univ. Press, New York 1993.
    • (1993) Riemannian geometry
    • Willmore, T.J.1
  • 30
    • 0035622710 scopus 로고    scopus 로고
    • Quaternionic holomorphic geometry: Plücker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori
    • D. Ferus, K. Leschke, P. Pedit, and U. Pinkall, "Quaternionic holomorphic geometry: Plücker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori", Invent. Math. 146:3 (2001), 507-593.
    • (2001) Invent. Math , vol.146 , Issue.3 , pp. 507-593
    • Ferus, D.1    Leschke, K.2    Pedit, P.3    Pinkall, U.4
  • 31
    • 85164640964 scopus 로고    scopus 로고
    • Russian source
    • Russian source
  • 32
    • 33746366182 scopus 로고    scopus 로고
    • Two-dimensional Dirac operator and the theory of surfaces
    • English transl., I. A. Taimanov, "Two-dimensional Dirac operator and the theory of surfaces", Russian Math. Surveys 61:1 (2006), 79-159.
    • (2006) Russian Math. Surveys , vol.61 , Issue.1 , pp. 79-159
    • English transl, I.1    Taimanov, A.2
  • 33
    • 0034392120 scopus 로고    scopus 로고
    • Lie sphere geometry and integrable systems
    • E.V. Ferapontov, "Lie sphere geometry and integrable systems", Tohôku Math. J. (2) 52:2 (2000), 199-233.
    • (2000) Tohôku Math. J. (2) , vol.52 , Issue.2 , pp. 199-233
    • Ferapontov, E.V.1
  • 34
    • 0036858789 scopus 로고    scopus 로고
    • Analog of Wilczynski's projective frame in Lie sphere geometry: Lie-applicable surfaces and commuting Schrödinger operators with magnetic fields
    • E. V. Ferapontov, "Analog of Wilczynski's projective frame in Lie sphere geometry: Lie-applicable surfaces and commuting Schrödinger operators with magnetic fields", Internat. J. Math. 13:9 (2002), 959-985.
    • (2002) Internat. J. Math , vol.13 , Issue.9 , pp. 959-985
    • Ferapontov, E.V.1
  • 35
    • 28844450270 scopus 로고    scopus 로고
    • On the Cauchy problem for the integrable system of Lie minimal surfaces
    • 11 2005, 113509, 15 p
    • E. Musso and L. Nicolodi, "On the Cauchy problem for the integrable system of Lie minimal surfaces", J. Math. Phys. 46:11 (2005), 113509, 15 p.
    • J. Math. Phys , vol.46
    • Musso, E.1    Nicolodi, L.2
  • 36
    • 0036988873 scopus 로고    scopus 로고
    • Harmonic maps in unfashionable geometries
    • F. Burstall and U. Hertrich-Jeromin, "Harmonic maps in unfashionable geometries", Manuscripta Math. 108:2 (2002), 171-189.
    • (2002) Manuscripta Math , vol.108 , Issue.2 , pp. 171-189
    • Burstall, F.1    Hertrich-Jeromin, U.2
  • 37
    • 33747366692 scopus 로고    scopus 로고
    • The Ribaucour transformation in Lie sphere geometry
    • F. E. Burstall and U. Hertrich-Jeromin, "The Ribaucour transformation in Lie sphere geometry", Differential Geom. Appl. 24:5 (2006), 503-520.
    • (2006) Differential Geom. Appl , vol.24 , Issue.5 , pp. 503-520
    • Burstall, F.E.1    Hertrich-Jeromin, U.2
  • 43
    • 0345702973 scopus 로고
    • Le transformazioni di Ribaucour dei sistemi n-pli ortogonali e il teorema generale di permutabilità
    • L. Bianchi, "Le transformazioni di Ribaucour dei sistemi n-pli ortogonali e il teorema generale di permutabilità", Annali di Mat. (3) 27 (1918), 183-257.
    • (1918) Annali di Mat , vol.27 , Issue.3 , pp. 183-257
    • Bianchi, L.1
  • 44
    • 33746348346 scopus 로고    scopus 로고
    • Introduction to Möbius differential geometry
    • London Math. Soc, Cambridge Univ. Press, Cambridge
    • U. Hertrich-Jeromin, Introduction to Möbius differential geometry, London Math. Soc. Lecture Note Ser., vol. 300, Cambridge Univ. Press, Cambridge 2003.
    • (2003) Lecture Note Ser , vol.300
    • Hertrich-Jeromin, U.1
  • 45
    • 85164638449 scopus 로고    scopus 로고
    • C. Rogers and W. K. Schief, Bäcklund and Darboux transformations. Geometry and modern applications in soliton theory, Cambridge Texts Appl. Math., Cambridge Univ. Press, Cambridge 2002.
    • C. Rogers and W. K. Schief, Bäcklund and Darboux transformations. Geometry and modern applications in soliton theory, Cambridge Texts Appl. Math., Cambridge Univ. Press, Cambridge 2002.
  • 46
    • 85164635613 scopus 로고    scopus 로고
    • Russian source
    • Russian source
  • 47
    • 0030327959 scopus 로고    scopus 로고
    • An algebraic formula for superposition and the completeness of the Backlund transformations of (2 + 1)-dimensional integrable systems
    • English transl., E. I. Ganzha and S. P. Tsarev, "An algebraic formula for superposition and the completeness of the Backlund transformations of (2 + 1)-dimensional integrable systems". Russian Math. Surveys 51:6 (1996), 1200-1202.
    • (1996) Russian Math. Surveys , vol.51 , Issue.6 , pp. 1200-1202
    • English transl, E.1    Ganzha, I.2    Tsarev, S.P.3
  • 48
    • 0003022531 scopus 로고    scopus 로고
    • Description of the n-orthogonal curvilinear coordinate systems and Hamiltonian integrable systems of hydrodynamic type. I: Integration of the Lamé equations
    • V. E. Zakharov, "Description of the n-orthogonal curvilinear coordinate systems and Hamiltonian integrable systems of hydrodynamic type. I: Integration of the Lamé equations", Duke Math. J. 94:1 (1998), 103-139.
    • (1998) Duke Math. J , vol.94 , Issue.1 , pp. 103-139
    • Zakharov, V.E.1
  • 49
    • 85164646423 scopus 로고    scopus 로고
    • Russian source
    • Russian source
  • 50
    • 0038439840 scopus 로고    scopus 로고
    • Algebraic-geometric n-orthogonal curvilinear coordinate systems and solutions of the associativity equations
    • English transl., I. M. Krichever, "Algebraic-geometric n-orthogonal curvilinear coordinate systems and solutions of the associativity equations", Funct. Anal. Appl. 31:1 (1997), 25-39.
    • (1997) Funct. Anal. Appl , vol.31 , Issue.1 , pp. 25-39
    • English transl, I.1    Krichever, M.2
  • 51
    • 0347427534 scopus 로고
    • Integrable systems in topological field theory
    • B. Dubrovin, "Integrable systems in topological field theory", Nuclear Phys. B 379:3 (1992), 627-689.
    • (1992) Nuclear Phys. B , vol.379 , Issue.3 , pp. 627-689
    • Dubrovin, B.1
  • 52
    • 0003533540 scopus 로고
    • Differential geometry applied to curve and surface design
    • Ellis Horwood Ltd, Chichester & Halsted Press, New York
    • A. W. Nutbourne and R. R. Martin, Differential geometry applied to curve and surface design, vol. 1: Foundations, Ellis Horwood Ltd., Chichester & Halsted Press, New York 1988.
    • (1988) Foundations , vol.1
    • Nutbourne, A.W.1    Martin, R.R.2
  • 53
    • 85164649471 scopus 로고    scopus 로고
    • A. I. Bobenko, Discrete conformal maps and surfaces, Symmetries and integrability of difference equations (Canterbury, 1996), London Math. Soc. Lecture Note Ser., 255, Cambridge Univ. Press, Cambridge 1999, pp. 97-108.
    • A. I. Bobenko, "Discrete conformal maps and surfaces", Symmetries and integrability of difference equations (Canterbury, 1996), London Math. Soc. Lecture Note Ser., vol. 255, Cambridge Univ. Press, Cambridge 1999, pp. 97-108.
  • 59
    • 0003649603 scopus 로고
    • I, Universitext, Springer-Verlag, Berlin
    • English transl., Geometry. I, Universitext, Springer-Verlag, Berlin 1987, 1994;
    • (1987) Geometry
    • transl, E.1
  • 60
    • 0003649603 scopus 로고
    • II, Universitext, Springer-Verlag, Berlin
    • Geometry. II, Universitext, Springer-Verlag, Berlin 1987.
    • (1987) Geometry
  • 62
    • 33746456381 scopus 로고
    • Three-dimensional integrable lattices in Euclidean spaces: Conjugacy and orthogonality
    • B. G. Konopelchenko and W. K. Schief, "Three-dimensional integrable lattices in Euclidean spaces: conjugacy and orthogonality", Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 454:1980 (1998), 3075-3104.
    • (1980) Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci , vol.454 , pp. 3075-3104
    • Konopelchenko, B.G.1    Schief, W.K.2
  • 63
    • 85164647125 scopus 로고    scopus 로고
    • A. Doliwa, S. V. Manakov, and P. M. Santini, ∂̄- reductions of the multidimensional quadrilateral lattice. The multidimensional circular lattice, Comm. Math. Phys. 196:1 (1998), 1-18.
    • A. Doliwa, S. V. Manakov, and P. M. Santini, "∂̄- reductions of the multidimensional quadrilateral lattice. The multidimensional circular lattice", Comm. Math. Phys. 196:1 (1998), 1-18.
  • 64
    • 85164646481 scopus 로고    scopus 로고
    • Russian source
    • Russian source
  • 67
    • 1542616797 scopus 로고    scopus 로고
    • On the unification of classical and novel integrable surfaces. II: Difference geometry
    • W. K. Schief, "On the unification of classical and novel integrable surfaces. II: Difference geometry", Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 459:2030 (2003), 373-391.
    • (2003) Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci , vol.459 , Issue.2030 , pp. 373-391
    • Schief, W.K.1
  • 68
    • 33745458246 scopus 로고    scopus 로고
    • On a maximum principle for minimal surfaces and their integrable discrete counterparts
    • W. K. Schief, "On a maximum principle for minimal surfaces and their integrable discrete counterparts", J. Geom. Phys. 56:9 (2006), 1484-1495.
    • (2006) J. Geom. Phys , vol.56 , Issue.9 , pp. 1484-1495
    • Schief, W.K.1
  • 69
    • 0034342507 scopus 로고    scopus 로고
    • Transformations of quadrilateral lattices
    • A. Doliwa, P.M. Santini, and M. Mañas, "Transformations of quadrilateral lattices", J. Math. Phys. 41:2 (2000), 944-990.
    • (2000) J. Math. Phys , vol.41 , Issue.2 , pp. 944-990
    • Doliwa, A.1    Santini, P.M.2    Mañas, M.3
  • 70
    • 0038785830 scopus 로고    scopus 로고
    • Discrete asymptotic nets and W-congruences in Plücker line geometry
    • A. Doliwa, "Discrete asymptotic nets and W-congruences in Plücker line geometry", J. Geom. Phys. 39:1 (2001), 9-29.
    • (2001) J. Geom. Phys , vol.39 , Issue.1 , pp. 9-29
    • Doliwa, A.1
  • 72
    • 34547469186 scopus 로고
    • 3rd ed, Birkhäuser, Basel-Stuttgart
    • W. Blaschke, Projektive Geometrie, 3rd ed., Birkhäuser, Basel-Stuttgart 1954.
    • (1954) Projektive Geometrie
    • Blaschke, W.1
  • 75
    • 0031999185 scopus 로고    scopus 로고
    • Applications of the Laguerre geometry in CAGD
    • H. Pottmann and M. Peternell, "Applications of the Laguerre geometry in CAGD", Comput. Aided Geom. Design 15:2 (1998), 165-186.
    • (1998) Comput. Aided Geom. Design , vol.15 , Issue.2 , pp. 165-186
    • Pottmann, H.1    Peternell, M.2
  • 76
    • 85164657105 scopus 로고    scopus 로고
    • A. Doliwa, The Ribaucour congruences of spheres within Lie sphere geometry, Bäcklund and Darboux transformations. The geometry of solutions (Halifax, NS, 1999), CRM Proc. Lecture Notes, 29, Amer. Math. Soc., Providence, RI 2001, pp. 159-166.
    • A. Doliwa, "The Ribaucour congruences of spheres within Lie sphere geometry", Bäcklund and Darboux transformations. The geometry of solutions (Halifax, NS, 1999), CRM Proc. Lecture Notes, vol. 29, Amer. Math. Soc., Providence, RI 2001, pp. 159-166.


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